Typo in Cafferelli-Silvestre?

Question

I am reading the 2008 paper by Caffarelli and Silvestre on Regularity theory for fully nonlinear integro-differential equations, and something is puzzling me.

On the third page, an operator $L$ is defined by \begin{equation} Lu(x)=\frac{1}{2}\int_{\mathbb{R}^n}(u(x+y)+u(x-y)-2u(x))K(y)dy, \end{equation}and the authors claim this is well-defined once we have $u\in C^{1,1}$ and $K$ satisfies \begin{equation} \int\frac{|y|^2}{1+|y|^2}K(y)dy<\infty. \end{equation}

However, just by taking $u(x)=x^2$ it is clear the convergence of the above integral does not necessary imply the well-definedness of $L$.

So is there a typo in the paper, or did I do something wrong?

Thanks!

Answer 1

On page $1$, the authors say:

The value of $Lu(x)$ is well defined as long as $u$ is bounded in $\mathbb R^n$ and $C^{ 1,1}$ at $x$.

The boundedness assumption is not stated for a few pages after that, but reappears in Definition 3.1.